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arXiv · 1909.09881

Applications of Forcing Theory to Homeomorphisms of the Closed Annulus

Jonathan Conejeros·Fabio Armando Tal
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Abstract

This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $μ$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measure $μ$ cannot be an endpoint of this segment. We also study the case of homeomorphisms such that $\mathbb{A}=(\mathbb{R}/\mathbb{Z})\times (0,1)$ is a region of instability of $f$. We show that, if the rotation numbers of the restriction of $f$ to the boundary components lies in the interior of the rotation set of $f$, then $f$ has uniformly bounded deviations from its rotation set. Finally, by combining this last result and recent work on realization of rotation vectors for annular continua, we obtain that if $f$ is any area-preserving homeomorphism of $\overline{\mathbb{A}}$ isotopic to the identity, then for every real number $ρ$ in the rotation set of $f$, there exists an associated Aubry-Mather set, that is, a compact $f$-invariant set such that every point in this set has a rotation number equal to $ρ$. This extends a result by P. Le Calvez previously known only for diffeomorphisms.

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BibTeXRIS

Jonathan Conejeros, Fabio Armando Tal. 2019-09-21. Applications of Forcing Theory to Homeomorphisms of the Closed Annulus. https://arxiv.org/abs/1909.09881

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